Integral

Struggling to show the factorial part of the expression, is my working out in showing adequate or is there a cleaner way to go about showing it?

$\displaystyle \ln(1^1 \cdot 2^2 \cdot 3^3 \cdots (n-1)^{n-1} \cdot n^n) = \sum_{i=1}^n i \ln(i)$

So work from there.

To add a semi-hint. It is not "obvious", but it's not actually that hard to get the LHS to look like the $\ln\left(\dfrac{n!}{0!}\right) + ... + \ln\left(\dfrac{n!}{0!}\right)$ expression.

Bit confused by what you mean to factor out, are you referring to
1x2x3...(n-1)xn = n!
1x2x3...(n-1)xn = n!/1!
3x4...(n-1)xn = n!/2!
...

hence the product of the above is ln (n! x n!/1! x n!/2! ... n!/(n-1)! ) which can be split into ln (n!/0! ) +
ln ( n!/1!) ... ln (n!/(n-1)!)

but this is essentially what I've already done in the above post if I made it a bit more clearer?

I have to be honest, I hadn't looked that far down - I was trusting when mqb said "top right seems to be along the right lines".

So, have you shown the result - um, maybe? You haven't actually "closed the gap"; you haven't even explictly said "this equals the expression in square brackets" (or words to that effect).

I'll be blunt: you really need to think a bit more about how you want to present your work. You're consistently on that fine line where an examiner might be able to work out what you meant and if it you actually knew how to prove the result, but they might easily decide "actually, with the amount of gaps I'm filling in myself I'm basically proving this for Student 999 - that's not OK".

It's often also just hard to puzzle out what you mean: the stuff you write e to work out what you meant, but it's going to take them ages, and if you're unlucky they're just not going to bother. I mean, look at your first line:

$\displaystyle \dfrac{x_n \ln\left(\frac{1}{x_n}\right)} {n} \text{ where } n\ge 0 \text{ where }x_0 = 0$

My immediate reaction is WTF is x_n? And even if I knew what x_n is, why have you written this expression down? And why has a sum appeared in the next line?

Oh God. I've just realised - you actually wrote this as a LH column going down to the bottom of the page, then restarted in the RH even though your LH column is (a lot) wider than where you started your RH column. I've been reading this in the wrong frickin' order.

So I think I'm going to stop at that point - obviously my initial comment about "closing the gap" is invalid. But the fact that a seasoned mathematician can look at your post and not be able to tell the correct order in which you've laid out your argument speaks for itself.

Appreciate the critique, I believe my post has came across in the wrong light due to my poor choice of words. I understand that presentation is lacking ,what I was really trying to ask was if the above that I typed up plus some kind of sentence explaining what I had done be sufficient to show the expression in the square brackets or is there some kind of thing they're looking for specifically like an induction proof which is more vigorous I guess.

the x_n thing didn't really make sense, I kind of wrote it out similar to when I did Simpson's rule where they use similar notation and just assumed anyone reading it would understand however will start explaining more of what I do. A better choice of notation would just be a sum sign.

Was too laxed in writing things out last night even though I do have 'unlimited paper' since its an iPad hopefully by the end of summer you'll see an improvement in the way I present things mathematically, in the meantime please do continue assisting me when you have time, thanks

you're consistently on that fine line where an examiner might be able to work out what you meant and if it you actually knew how to prove the result, but they might easily decide "actually, with the amount of gaps I'm filling in myself I'm basically proving this for Student 999 - that's not OK"